To\"en's formula and Green's formula

TL;DR

This paper provides a simplified proof of Green's formula for Hall algebras by leveraging the associativity of the derived Hall algebra, which is based on Toën's formula, in the context of finitary hereditary abelian categories.

Contribution

It introduces a straightforward proof of Green's formula using the associativity of the derived Hall algebra linked to Toën's formula.

Findings

Established a simple proof of Green's formula.

Connected Toën's formula with the associativity of derived Hall algebras.

Clarified the algebraic structure underlying Hall algebras.

Abstract

Let $\mathcal {A}$ be a finitary hereditary abelian category. In this note, we use the associativity of the derived Hall algebra associated to the bounded derived category of $\mathcal {A}$, whose multiplication structure constants are given by the so-called To\"en's formula, to give a simple proof of Green's formula associated to Hall algebras.